The Central Tendency, Weighted Likelihood, and Exponential family

In this paper, we establish the links between the H\"older and Lehmer central tendencies and the maximum likelihood for the estimation of the one-parameter exponential family of probability density functions. For this, we show that the maximum weighted likelihood of the parameter is a generalized weighted mean from which the central tendencies of H\"older and Lehmer can be inferred. Some of the links obtained do not seem to be part of the state of the art. Moreover, we show that the maximum weighted likelihood is equivalent to the minimum of the weighted least square error. Experimentations confirm that the maximum weighted likelihood leads to a more accurate fitting of histograms.